3.13.99 \(\int (A+B x) (d+e x)^m (a+c x^2)^2 \, dx\)
Optimal. Leaf size=234 \[ -\frac {\left (a e^2+c d^2\right )^2 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac {\left (a e^2+c d^2\right ) (d+e x)^{m+2} \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 (m+2)}+\frac {2 c (d+e x)^{m+4} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (m+4)}-\frac {2 c (d+e x)^{m+3} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (m+3)}-\frac {c^2 (5 B d-A e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac {B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]
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Rubi [A] time = 0.14, antiderivative size = 234, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used =
{772} \begin {gather*} -\frac {\left (a e^2+c d^2\right )^2 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac {\left (a e^2+c d^2\right ) (d+e x)^{m+2} \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 (m+2)}-\frac {2 c (d+e x)^{m+3} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (m+3)}+\frac {2 c (d+e x)^{m+4} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (m+4)}-\frac {c^2 (5 B d-A e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac {B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^m*(a + c*x^2)^2,x]
[Out]
-(((B*d - A*e)*(c*d^2 + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + ((c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e +
a*B*e^2)*(d + e*x)^(2 + m))/(e^6*(2 + m)) - (2*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^
(3 + m))/(e^6*(3 + m)) + (2*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (c^2*(5*B*d
- A*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6 + m))
Rule 772
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2 (d+e x)^m}{e^5}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^{1+m}}{e^5}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^{2+m}}{e^5}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^{3+m}}{e^5}+\frac {c^2 (-5 B d+A e) (d+e x)^{4+m}}{e^5}+\frac {B c^2 (d+e x)^{5+m}}{e^5}\right ) \, dx\\ &=-\frac {(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^{1+m}}{e^6 (1+m)}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^{2+m}}{e^6 (2+m)}-\frac {2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{3+m}}{e^6 (3+m)}+\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac {c^2 (5 B d-A e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {B c^2 (d+e x)^{6+m}}{e^6 (6+m)}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 355, normalized size = 1.52 \begin {gather*} \frac {(d+e x)^{m+1} \left (B (m+1) (d+e x) \left (4 (m+5) \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+7 m+12\right )+c \left (2 d^2-2 d e (m+2) x+e^2 \left (m^2+5 m+6\right ) x^2\right )\right )-4 c d (m+2) (d+e x) \left (a e^2 \left (m^2+9 m+20\right )+c \left (2 d^2-2 d e (m+3) x+e^2 \left (m^2+7 m+12\right ) x^2\right )\right )+e^4 (m+2) (m+3) (m+4) (m+5) \left (a+c x^2\right )^2\right )-(m+6) (B d-A e) \left (4 (m+4) \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+5 m+6\right )+c \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )-4 c d (m+1) (d+e x) \left (a e^2 \left (m^2+7 m+12\right )+c \left (2 d^2-2 d e (m+2) x+e^2 \left (m^2+5 m+6\right ) x^2\right )\right )+e^4 (m+1) (m+2) (m+3) (m+4) \left (a+c x^2\right )^2\right )\right )}{e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^m*(a + c*x^2)^2,x]
[Out]
((d + e*x)^(1 + m)*(-((B*d - A*e)*(6 + m)*(e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^
2)*(4 + m)*(a*e^2*(6 + 5*m + m^2) + c*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)) - 4*c*d*(1 + m)*(d
+ e*x)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)))) + B*(1 + m)*(d + e*x
)*(e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)*(5 + m)*(a*e^2*(12 + 7*m + m^2) + c*(
2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)) - 4*c*d*(2 + m)*(d + e*x)*(a*e^2*(20 + 9*m + m^2) + c*(2*d
^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7*m + m^2)*x^2)))))/(e^6*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m))
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IntegrateAlgebraic [F] time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^m \left (a+c x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(A + B*x)*(d + e*x)^m*(a + c*x^2)^2,x]
[Out]
Defer[IntegrateAlgebraic][(A + B*x)*(d + e*x)^m*(a + c*x^2)^2, x]
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fricas [B] time = 0.44, size = 1373, normalized size = 5.87
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(c*x^2+a)^2,x, algorithm="fricas")
[Out]
(A*a^2*d*e^5*m^5 - 120*B*c^2*d^6 + 144*A*c^2*d^5*e - 360*B*a*c*d^4*e^2 + 480*A*a*c*d^3*e^3 - 360*B*a^2*d^2*e^4
+ 720*A*a^2*d*e^5 + (B*c^2*e^6*m^5 + 15*B*c^2*e^6*m^4 + 85*B*c^2*e^6*m^3 + 225*B*c^2*e^6*m^2 + 274*B*c^2*e^6*
m + 120*B*c^2*e^6)*x^6 + (144*A*c^2*e^6 + (B*c^2*d*e^5 + A*c^2*e^6)*m^5 + 2*(5*B*c^2*d*e^5 + 8*A*c^2*e^6)*m^4
+ 5*(7*B*c^2*d*e^5 + 19*A*c^2*e^6)*m^3 + 10*(5*B*c^2*d*e^5 + 26*A*c^2*e^6)*m^2 + 12*(2*B*c^2*d*e^5 + 27*A*c^2*
e^6)*m)*x^5 - (B*a^2*d^2*e^4 - 20*A*a^2*d*e^5)*m^4 + (360*B*a*c*e^6 + (A*c^2*d*e^5 + 2*B*a*c*e^6)*m^5 - (5*B*c
^2*d^2*e^4 - 12*A*c^2*d*e^5 - 34*B*a*c*e^6)*m^4 - (30*B*c^2*d^2*e^4 - 47*A*c^2*d*e^5 - 214*B*a*c*e^6)*m^3 - (5
5*B*c^2*d^2*e^4 - 72*A*c^2*d*e^5 - 614*B*a*c*e^6)*m^2 - 6*(5*B*c^2*d^2*e^4 - 6*A*c^2*d*e^5 - 132*B*a*c*e^6)*m)
*x^4 + (4*A*a*c*d^3*e^3 - 18*B*a^2*d^2*e^4 + 155*A*a^2*d*e^5)*m^3 + 2*(240*A*a*c*e^6 + (B*a*c*d*e^5 + A*a*c*e^
6)*m^5 - 2*(A*c^2*d^2*e^4 - 7*B*a*c*d*e^5 - 9*A*a*c*e^6)*m^4 + (10*B*c^2*d^3*e^3 - 18*A*c^2*d^2*e^4 + 65*B*a*c
*d*e^5 + 121*A*a*c*e^6)*m^3 + 2*(15*B*c^2*d^3*e^3 - 20*A*c^2*d^2*e^4 + 56*B*a*c*d*e^5 + 186*A*a*c*e^6)*m^2 + 4
*(5*B*c^2*d^3*e^3 - 6*A*c^2*d^2*e^4 + 15*B*a*c*d*e^5 + 127*A*a*c*e^6)*m)*x^3 - (12*B*a*c*d^4*e^2 - 60*A*a*c*d^
3*e^3 + 119*B*a^2*d^2*e^4 - 580*A*a^2*d*e^5)*m^2 + (360*B*a^2*e^6 + (2*A*a*c*d*e^5 + B*a^2*e^6)*m^5 - (6*B*a*c
*d^2*e^4 - 32*A*a*c*d*e^5 - 19*B*a^2*e^6)*m^4 + (12*A*c^2*d^3*e^3 - 72*B*a*c*d^2*e^4 + 178*A*a*c*d*e^5 + 137*B
*a^2*e^6)*m^3 - (60*B*c^2*d^4*e^2 - 84*A*c^2*d^3*e^3 + 246*B*a*c*d^2*e^4 - 388*A*a*c*d*e^5 - 461*B*a^2*e^6)*m^
2 - 6*(10*B*c^2*d^4*e^2 - 12*A*c^2*d^3*e^3 + 30*B*a*c*d^2*e^4 - 40*A*a*c*d*e^5 - 117*B*a^2*e^6)*m)*x^2 + 2*(12
*A*c^2*d^5*e - 66*B*a*c*d^4*e^2 + 148*A*a*c*d^3*e^3 - 171*B*a^2*d^2*e^4 + 522*A*a^2*d*e^5)*m + (720*A*a^2*e^6
+ (B*a^2*d*e^5 + A*a^2*e^6)*m^5 - 2*(2*A*a*c*d^2*e^4 - 9*B*a^2*d*e^5 - 10*A*a^2*e^6)*m^4 + (12*B*a*c*d^3*e^3 -
60*A*a*c*d^2*e^4 + 119*B*a^2*d*e^5 + 155*A*a^2*e^6)*m^3 - 2*(12*A*c^2*d^4*e^2 - 66*B*a*c*d^3*e^3 + 148*A*a*c*
d^2*e^4 - 171*B*a^2*d*e^5 - 290*A*a^2*e^6)*m^2 + 12*(10*B*c^2*d^5*e - 12*A*c^2*d^4*e^2 + 30*B*a*c*d^3*e^3 - 40
*A*a*c*d^2*e^4 + 30*B*a^2*d*e^5 + 87*A*a^2*e^6)*m)*x)*(e*x + d)^m/(e^6*m^6 + 21*e^6*m^5 + 175*e^6*m^4 + 735*e^
6*m^3 + 1624*e^6*m^2 + 1764*e^6*m + 720*e^6)
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giac [B] time = 0.26, size = 2435, normalized size = 10.41
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(c*x^2+a)^2,x, algorithm="giac")
[Out]
((x*e + d)^m*B*c^2*m^5*x^6*e^6 + (x*e + d)^m*B*c^2*d*m^5*x^5*e^5 + (x*e + d)^m*A*c^2*m^5*x^5*e^6 + 15*(x*e + d
)^m*B*c^2*m^4*x^6*e^6 + (x*e + d)^m*A*c^2*d*m^5*x^4*e^5 + 10*(x*e + d)^m*B*c^2*d*m^4*x^5*e^5 - 5*(x*e + d)^m*B
*c^2*d^2*m^4*x^4*e^4 + 2*(x*e + d)^m*B*a*c*m^5*x^4*e^6 + 16*(x*e + d)^m*A*c^2*m^4*x^5*e^6 + 85*(x*e + d)^m*B*c
^2*m^3*x^6*e^6 + 2*(x*e + d)^m*B*a*c*d*m^5*x^3*e^5 + 12*(x*e + d)^m*A*c^2*d*m^4*x^4*e^5 + 35*(x*e + d)^m*B*c^2
*d*m^3*x^5*e^5 - 4*(x*e + d)^m*A*c^2*d^2*m^4*x^3*e^4 - 30*(x*e + d)^m*B*c^2*d^2*m^3*x^4*e^4 + 20*(x*e + d)^m*B
*c^2*d^3*m^3*x^3*e^3 + 2*(x*e + d)^m*A*a*c*m^5*x^3*e^6 + 34*(x*e + d)^m*B*a*c*m^4*x^4*e^6 + 95*(x*e + d)^m*A*c
^2*m^3*x^5*e^6 + 225*(x*e + d)^m*B*c^2*m^2*x^6*e^6 + 2*(x*e + d)^m*A*a*c*d*m^5*x^2*e^5 + 28*(x*e + d)^m*B*a*c*
d*m^4*x^3*e^5 + 47*(x*e + d)^m*A*c^2*d*m^3*x^4*e^5 + 50*(x*e + d)^m*B*c^2*d*m^2*x^5*e^5 - 6*(x*e + d)^m*B*a*c*
d^2*m^4*x^2*e^4 - 36*(x*e + d)^m*A*c^2*d^2*m^3*x^3*e^4 - 55*(x*e + d)^m*B*c^2*d^2*m^2*x^4*e^4 + 12*(x*e + d)^m
*A*c^2*d^3*m^3*x^2*e^3 + 60*(x*e + d)^m*B*c^2*d^3*m^2*x^3*e^3 - 60*(x*e + d)^m*B*c^2*d^4*m^2*x^2*e^2 + (x*e +
d)^m*B*a^2*m^5*x^2*e^6 + 36*(x*e + d)^m*A*a*c*m^4*x^3*e^6 + 214*(x*e + d)^m*B*a*c*m^3*x^4*e^6 + 260*(x*e + d)^
m*A*c^2*m^2*x^5*e^6 + 274*(x*e + d)^m*B*c^2*m*x^6*e^6 + (x*e + d)^m*B*a^2*d*m^5*x*e^5 + 32*(x*e + d)^m*A*a*c*d
*m^4*x^2*e^5 + 130*(x*e + d)^m*B*a*c*d*m^3*x^3*e^5 + 72*(x*e + d)^m*A*c^2*d*m^2*x^4*e^5 + 24*(x*e + d)^m*B*c^2
*d*m*x^5*e^5 - 4*(x*e + d)^m*A*a*c*d^2*m^4*x*e^4 - 72*(x*e + d)^m*B*a*c*d^2*m^3*x^2*e^4 - 80*(x*e + d)^m*A*c^2
*d^2*m^2*x^3*e^4 - 30*(x*e + d)^m*B*c^2*d^2*m*x^4*e^4 + 12*(x*e + d)^m*B*a*c*d^3*m^3*x*e^3 + 84*(x*e + d)^m*A*
c^2*d^3*m^2*x^2*e^3 + 40*(x*e + d)^m*B*c^2*d^3*m*x^3*e^3 - 24*(x*e + d)^m*A*c^2*d^4*m^2*x*e^2 - 60*(x*e + d)^m
*B*c^2*d^4*m*x^2*e^2 + 120*(x*e + d)^m*B*c^2*d^5*m*x*e + (x*e + d)^m*A*a^2*m^5*x*e^6 + 19*(x*e + d)^m*B*a^2*m^
4*x^2*e^6 + 242*(x*e + d)^m*A*a*c*m^3*x^3*e^6 + 614*(x*e + d)^m*B*a*c*m^2*x^4*e^6 + 324*(x*e + d)^m*A*c^2*m*x^
5*e^6 + 120*(x*e + d)^m*B*c^2*x^6*e^6 + (x*e + d)^m*A*a^2*d*m^5*e^5 + 18*(x*e + d)^m*B*a^2*d*m^4*x*e^5 + 178*(
x*e + d)^m*A*a*c*d*m^3*x^2*e^5 + 224*(x*e + d)^m*B*a*c*d*m^2*x^3*e^5 + 36*(x*e + d)^m*A*c^2*d*m*x^4*e^5 - (x*e
+ d)^m*B*a^2*d^2*m^4*e^4 - 60*(x*e + d)^m*A*a*c*d^2*m^3*x*e^4 - 246*(x*e + d)^m*B*a*c*d^2*m^2*x^2*e^4 - 48*(x
*e + d)^m*A*c^2*d^2*m*x^3*e^4 + 4*(x*e + d)^m*A*a*c*d^3*m^3*e^3 + 132*(x*e + d)^m*B*a*c*d^3*m^2*x*e^3 + 72*(x*
e + d)^m*A*c^2*d^3*m*x^2*e^3 - 12*(x*e + d)^m*B*a*c*d^4*m^2*e^2 - 144*(x*e + d)^m*A*c^2*d^4*m*x*e^2 + 24*(x*e
+ d)^m*A*c^2*d^5*m*e - 120*(x*e + d)^m*B*c^2*d^6 + 20*(x*e + d)^m*A*a^2*m^4*x*e^6 + 137*(x*e + d)^m*B*a^2*m^3*
x^2*e^6 + 744*(x*e + d)^m*A*a*c*m^2*x^3*e^6 + 792*(x*e + d)^m*B*a*c*m*x^4*e^6 + 144*(x*e + d)^m*A*c^2*x^5*e^6
+ 20*(x*e + d)^m*A*a^2*d*m^4*e^5 + 119*(x*e + d)^m*B*a^2*d*m^3*x*e^5 + 388*(x*e + d)^m*A*a*c*d*m^2*x^2*e^5 + 1
20*(x*e + d)^m*B*a*c*d*m*x^3*e^5 - 18*(x*e + d)^m*B*a^2*d^2*m^3*e^4 - 296*(x*e + d)^m*A*a*c*d^2*m^2*x*e^4 - 18
0*(x*e + d)^m*B*a*c*d^2*m*x^2*e^4 + 60*(x*e + d)^m*A*a*c*d^3*m^2*e^3 + 360*(x*e + d)^m*B*a*c*d^3*m*x*e^3 - 132
*(x*e + d)^m*B*a*c*d^4*m*e^2 + 144*(x*e + d)^m*A*c^2*d^5*e + 155*(x*e + d)^m*A*a^2*m^3*x*e^6 + 461*(x*e + d)^m
*B*a^2*m^2*x^2*e^6 + 1016*(x*e + d)^m*A*a*c*m*x^3*e^6 + 360*(x*e + d)^m*B*a*c*x^4*e^6 + 155*(x*e + d)^m*A*a^2*
d*m^3*e^5 + 342*(x*e + d)^m*B*a^2*d*m^2*x*e^5 + 240*(x*e + d)^m*A*a*c*d*m*x^2*e^5 - 119*(x*e + d)^m*B*a^2*d^2*
m^2*e^4 - 480*(x*e + d)^m*A*a*c*d^2*m*x*e^4 + 296*(x*e + d)^m*A*a*c*d^3*m*e^3 - 360*(x*e + d)^m*B*a*c*d^4*e^2
+ 580*(x*e + d)^m*A*a^2*m^2*x*e^6 + 702*(x*e + d)^m*B*a^2*m*x^2*e^6 + 480*(x*e + d)^m*A*a*c*x^3*e^6 + 580*(x*e
+ d)^m*A*a^2*d*m^2*e^5 + 360*(x*e + d)^m*B*a^2*d*m*x*e^5 - 342*(x*e + d)^m*B*a^2*d^2*m*e^4 + 480*(x*e + d)^m*
A*a*c*d^3*e^3 + 1044*(x*e + d)^m*A*a^2*m*x*e^6 + 360*(x*e + d)^m*B*a^2*x^2*e^6 + 1044*(x*e + d)^m*A*a^2*d*m*e^
5 - 360*(x*e + d)^m*B*a^2*d^2*e^4 + 720*(x*e + d)^m*A*a^2*x*e^6 + 720*(x*e + d)^m*A*a^2*d*e^5)/(m^6*e^6 + 21*m
^5*e^6 + 175*m^4*e^6 + 735*m^3*e^6 + 1624*m^2*e^6 + 1764*m*e^6 + 720*e^6)
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maple [B] time = 0.06, size = 1271, normalized size = 5.43 \begin {gather*} \frac {\left (B \,c^{2} e^{5} m^{5} x^{5}+A \,c^{2} e^{5} m^{5} x^{4}+15 B \,c^{2} e^{5} m^{4} x^{5}+16 A \,c^{2} e^{5} m^{4} x^{4}+2 B a c \,e^{5} m^{5} x^{3}-5 B \,c^{2} d \,e^{4} m^{4} x^{4}+85 B \,c^{2} e^{5} m^{3} x^{5}+2 A a c \,e^{5} m^{5} x^{2}-4 A \,c^{2} d \,e^{4} m^{4} x^{3}+95 A \,c^{2} e^{5} m^{3} x^{4}+34 B a c \,e^{5} m^{4} x^{3}-50 B \,c^{2} d \,e^{4} m^{3} x^{4}+225 B \,c^{2} e^{5} m^{2} x^{5}+36 A a c \,e^{5} m^{4} x^{2}-48 A \,c^{2} d \,e^{4} m^{3} x^{3}+260 A \,c^{2} e^{5} m^{2} x^{4}+B \,a^{2} e^{5} m^{5} x -6 B a c d \,e^{4} m^{4} x^{2}+214 B a c \,e^{5} m^{3} x^{3}+20 B \,c^{2} d^{2} e^{3} m^{3} x^{3}-175 B \,c^{2} d \,e^{4} m^{2} x^{4}+274 B \,c^{2} e^{5} m \,x^{5}+A \,a^{2} e^{5} m^{5}-4 A a c d \,e^{4} m^{4} x +242 A a c \,e^{5} m^{3} x^{2}+12 A \,c^{2} d^{2} e^{3} m^{3} x^{2}-188 A \,c^{2} d \,e^{4} m^{2} x^{3}+324 A \,c^{2} e^{5} m \,x^{4}+19 B \,a^{2} e^{5} m^{4} x -84 B a c d \,e^{4} m^{3} x^{2}+614 B a c \,e^{5} m^{2} x^{3}+120 B \,c^{2} d^{2} e^{3} m^{2} x^{3}-250 B \,c^{2} d \,e^{4} m \,x^{4}+120 B \,x^{5} c^{2} e^{5}+20 A \,a^{2} e^{5} m^{4}-64 A a c d \,e^{4} m^{3} x +744 A a c \,e^{5} m^{2} x^{2}+108 A \,c^{2} d^{2} e^{3} m^{2} x^{2}-288 A \,c^{2} d \,e^{4} m \,x^{3}+144 A \,c^{2} e^{5} x^{4}-B \,a^{2} d \,e^{4} m^{4}+137 B \,a^{2} e^{5} m^{3} x +12 B a c \,d^{2} e^{3} m^{3} x -390 B a c d \,e^{4} m^{2} x^{2}+792 B a c \,e^{5} m \,x^{3}-60 B \,c^{2} d^{3} e^{2} m^{2} x^{2}+220 B \,c^{2} d^{2} e^{3} m \,x^{3}-120 B \,c^{2} d \,e^{4} x^{4}+155 A \,a^{2} e^{5} m^{3}+4 A a c \,d^{2} e^{3} m^{3}-356 A a c d \,e^{4} m^{2} x +1016 A a c \,e^{5} m \,x^{2}-24 A \,c^{2} d^{3} e^{2} m^{2} x +240 A \,c^{2} d^{2} e^{3} m \,x^{2}-144 A \,c^{2} d \,e^{4} x^{3}-18 B \,a^{2} d \,e^{4} m^{3}+461 B \,a^{2} e^{5} m^{2} x +144 B a c \,d^{2} e^{3} m^{2} x -672 B a c d \,e^{4} m \,x^{2}+360 B a c \,e^{5} x^{3}-180 B \,c^{2} d^{3} e^{2} m \,x^{2}+120 B \,c^{2} d^{2} e^{3} x^{3}+580 A \,a^{2} e^{5} m^{2}+60 A a c \,d^{2} e^{3} m^{2}-776 A a c d \,e^{4} m x +480 A a c \,e^{5} x^{2}-168 A \,c^{2} d^{3} e^{2} m x +144 A \,c^{2} d^{2} e^{3} x^{2}-119 B \,a^{2} d \,e^{4} m^{2}+702 B \,a^{2} e^{5} m x -12 B a c \,d^{3} e^{2} m^{2}+492 B a c \,d^{2} e^{3} m x -360 B a c d \,e^{4} x^{2}+120 B \,c^{2} d^{4} e m x -120 B \,c^{2} d^{3} e^{2} x^{2}+1044 A \,a^{2} e^{5} m +296 A a c \,d^{2} e^{3} m -480 A a c d \,e^{4} x +24 A \,c^{2} d^{4} e m -144 A \,c^{2} d^{3} e^{2} x -342 B \,a^{2} d \,e^{4} m +360 B x \,a^{2} e^{5}-132 B a c \,d^{3} e^{2} m +360 B a c \,d^{2} e^{3} x +120 B \,c^{2} d^{4} e x +720 A \,a^{2} e^{5}+480 A \,d^{2} a c \,e^{3}+144 A \,c^{2} d^{4} e -360 B \,a^{2} d \,e^{4}-360 B \,d^{3} a c \,e^{2}-120 B \,c^{2} d^{5}\right ) \left (e x +d \right )^{m +1}}{\left (m^{6}+21 m^{5}+175 m^{4}+735 m^{3}+1624 m^{2}+1764 m +720\right ) e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^m*(c*x^2+a)^2,x)
[Out]
(e*x+d)^(m+1)*(B*c^2*e^5*m^5*x^5+A*c^2*e^5*m^5*x^4+15*B*c^2*e^5*m^4*x^5+16*A*c^2*e^5*m^4*x^4+2*B*a*c*e^5*m^5*x
^3-5*B*c^2*d*e^4*m^4*x^4+85*B*c^2*e^5*m^3*x^5+2*A*a*c*e^5*m^5*x^2-4*A*c^2*d*e^4*m^4*x^3+95*A*c^2*e^5*m^3*x^4+3
4*B*a*c*e^5*m^4*x^3-50*B*c^2*d*e^4*m^3*x^4+225*B*c^2*e^5*m^2*x^5+36*A*a*c*e^5*m^4*x^2-48*A*c^2*d*e^4*m^3*x^3+2
60*A*c^2*e^5*m^2*x^4+B*a^2*e^5*m^5*x-6*B*a*c*d*e^4*m^4*x^2+214*B*a*c*e^5*m^3*x^3+20*B*c^2*d^2*e^3*m^3*x^3-175*
B*c^2*d*e^4*m^2*x^4+274*B*c^2*e^5*m*x^5+A*a^2*e^5*m^5-4*A*a*c*d*e^4*m^4*x+242*A*a*c*e^5*m^3*x^2+12*A*c^2*d^2*e
^3*m^3*x^2-188*A*c^2*d*e^4*m^2*x^3+324*A*c^2*e^5*m*x^4+19*B*a^2*e^5*m^4*x-84*B*a*c*d*e^4*m^3*x^2+614*B*a*c*e^5
*m^2*x^3+120*B*c^2*d^2*e^3*m^2*x^3-250*B*c^2*d*e^4*m*x^4+120*B*c^2*e^5*x^5+20*A*a^2*e^5*m^4-64*A*a*c*d*e^4*m^3
*x+744*A*a*c*e^5*m^2*x^2+108*A*c^2*d^2*e^3*m^2*x^2-288*A*c^2*d*e^4*m*x^3+144*A*c^2*e^5*x^4-B*a^2*d*e^4*m^4+137
*B*a^2*e^5*m^3*x+12*B*a*c*d^2*e^3*m^3*x-390*B*a*c*d*e^4*m^2*x^2+792*B*a*c*e^5*m*x^3-60*B*c^2*d^3*e^2*m^2*x^2+2
20*B*c^2*d^2*e^3*m*x^3-120*B*c^2*d*e^4*x^4+155*A*a^2*e^5*m^3+4*A*a*c*d^2*e^3*m^3-356*A*a*c*d*e^4*m^2*x+1016*A*
a*c*e^5*m*x^2-24*A*c^2*d^3*e^2*m^2*x+240*A*c^2*d^2*e^3*m*x^2-144*A*c^2*d*e^4*x^3-18*B*a^2*d*e^4*m^3+461*B*a^2*
e^5*m^2*x+144*B*a*c*d^2*e^3*m^2*x-672*B*a*c*d*e^4*m*x^2+360*B*a*c*e^5*x^3-180*B*c^2*d^3*e^2*m*x^2+120*B*c^2*d^
2*e^3*x^3+580*A*a^2*e^5*m^2+60*A*a*c*d^2*e^3*m^2-776*A*a*c*d*e^4*m*x+480*A*a*c*e^5*x^2-168*A*c^2*d^3*e^2*m*x+1
44*A*c^2*d^2*e^3*x^2-119*B*a^2*d*e^4*m^2+702*B*a^2*e^5*m*x-12*B*a*c*d^3*e^2*m^2+492*B*a*c*d^2*e^3*m*x-360*B*a*
c*d*e^4*x^2+120*B*c^2*d^4*e*m*x-120*B*c^2*d^3*e^2*x^2+1044*A*a^2*e^5*m+296*A*a*c*d^2*e^3*m-480*A*a*c*d*e^4*x+2
4*A*c^2*d^4*e*m-144*A*c^2*d^3*e^2*x-342*B*a^2*d*e^4*m+360*B*a^2*e^5*x-132*B*a*c*d^3*e^2*m+360*B*a*c*d^2*e^3*x+
120*B*c^2*d^4*e*x+720*A*a^2*e^5+480*A*a*c*d^2*e^3+144*A*c^2*d^4*e-360*B*a^2*d*e^4-360*B*a*c*d^3*e^2-120*B*c^2*
d^5)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+720)
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maxima [B] time = 0.74, size = 575, normalized size = 2.46 \begin {gather*} \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} B a^{2}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} A a^{2}}{e {\left (m + 1\right )}} + \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} A a c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} B a c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} A c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {{\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m} B c^{2}}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(c*x^2+a)^2,x, algorithm="maxima")
[Out]
(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*B*a^2/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*A*a^2/(e*(m + 1)
) + 2*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*A*a*c/((m^3 + 6*m^2 +
11*m + 6)*e^3) + 2*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2
+ 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*B*a*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m^2
+ 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 +
m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*A*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5
) + ((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x
^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^
2 + 120*d^5*e*m*x - 120*d^6)*(e*x + d)^m*B*c^2/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e
^6)
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mupad [B] time = 2.46, size = 1229, normalized size = 5.25 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (-B\,a^2\,d^2\,e^4\,m^4-18\,B\,a^2\,d^2\,e^4\,m^3-119\,B\,a^2\,d^2\,e^4\,m^2-342\,B\,a^2\,d^2\,e^4\,m-360\,B\,a^2\,d^2\,e^4+A\,a^2\,d\,e^5\,m^5+20\,A\,a^2\,d\,e^5\,m^4+155\,A\,a^2\,d\,e^5\,m^3+580\,A\,a^2\,d\,e^5\,m^2+1044\,A\,a^2\,d\,e^5\,m+720\,A\,a^2\,d\,e^5-12\,B\,a\,c\,d^4\,e^2\,m^2-132\,B\,a\,c\,d^4\,e^2\,m-360\,B\,a\,c\,d^4\,e^2+4\,A\,a\,c\,d^3\,e^3\,m^3+60\,A\,a\,c\,d^3\,e^3\,m^2+296\,A\,a\,c\,d^3\,e^3\,m+480\,A\,a\,c\,d^3\,e^3-120\,B\,c^2\,d^6+24\,A\,c^2\,d^5\,e\,m+144\,A\,c^2\,d^5\,e\right )}{e^6\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (B\,a^2\,d\,e^5\,m^5+18\,B\,a^2\,d\,e^5\,m^4+119\,B\,a^2\,d\,e^5\,m^3+342\,B\,a^2\,d\,e^5\,m^2+360\,B\,a^2\,d\,e^5\,m+A\,a^2\,e^6\,m^5+20\,A\,a^2\,e^6\,m^4+155\,A\,a^2\,e^6\,m^3+580\,A\,a^2\,e^6\,m^2+1044\,A\,a^2\,e^6\,m+720\,A\,a^2\,e^6+12\,B\,a\,c\,d^3\,e^3\,m^3+132\,B\,a\,c\,d^3\,e^3\,m^2+360\,B\,a\,c\,d^3\,e^3\,m-4\,A\,a\,c\,d^2\,e^4\,m^4-60\,A\,a\,c\,d^2\,e^4\,m^3-296\,A\,a\,c\,d^2\,e^4\,m^2-480\,A\,a\,c\,d^2\,e^4\,m+120\,B\,c^2\,d^5\,e\,m-24\,A\,c^2\,d^4\,e^2\,m^2-144\,A\,c^2\,d^4\,e^2\,m\right )}{e^6\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (B\,a^2\,e^4\,m^4+18\,B\,a^2\,e^4\,m^3+119\,B\,a^2\,e^4\,m^2+342\,B\,a^2\,e^4\,m+360\,B\,a^2\,e^4-6\,B\,a\,c\,d^2\,e^2\,m^3-66\,B\,a\,c\,d^2\,e^2\,m^2-180\,B\,a\,c\,d^2\,e^2\,m+2\,A\,a\,c\,d\,e^3\,m^4+30\,A\,a\,c\,d\,e^3\,m^3+148\,A\,a\,c\,d\,e^3\,m^2+240\,A\,a\,c\,d\,e^3\,m-60\,B\,c^2\,d^4\,m+12\,A\,c^2\,d^3\,e\,m^2+72\,A\,c^2\,d^3\,e\,m\right )}{e^4\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {B\,c^2\,x^6\,{\left (d+e\,x\right )}^m\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {c^2\,x^5\,{\left (d+e\,x\right )}^m\,\left (6\,A\,e+A\,e\,m+B\,d\,m\right )\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{e\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {c\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (-5\,B\,c\,d^2\,m+A\,c\,d\,e\,m^2+6\,A\,c\,d\,e\,m+2\,B\,a\,e^2\,m^2+22\,B\,a\,e^2\,m+60\,B\,a\,e^2\right )}{e^2\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {2\,c\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (10\,B\,c\,d^3\,m-2\,A\,c\,d^2\,e\,m^2-12\,A\,c\,d^2\,e\,m+B\,a\,d\,e^2\,m^3+11\,B\,a\,d\,e^2\,m^2+30\,B\,a\,d\,e^2\,m+A\,a\,e^3\,m^3+15\,A\,a\,e^3\,m^2+74\,A\,a\,e^3\,m+120\,A\,a\,e^3\right )}{e^3\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + c*x^2)^2*(A + B*x)*(d + e*x)^m,x)
[Out]
((d + e*x)^m*(720*A*a^2*d*e^5 - 120*B*c^2*d^6 + 144*A*c^2*d^5*e - 360*B*a^2*d^2*e^4 + 580*A*a^2*d*e^5*m^2 + 15
5*A*a^2*d*e^5*m^3 + 20*A*a^2*d*e^5*m^4 + A*a^2*d*e^5*m^5 - 342*B*a^2*d^2*e^4*m - 119*B*a^2*d^2*e^4*m^2 - 18*B*
a^2*d^2*e^4*m^3 - B*a^2*d^2*e^4*m^4 + 480*A*a*c*d^3*e^3 - 360*B*a*c*d^4*e^2 + 1044*A*a^2*d*e^5*m + 24*A*c^2*d^
5*e*m + 296*A*a*c*d^3*e^3*m - 132*B*a*c*d^4*e^2*m + 60*A*a*c*d^3*e^3*m^2 + 4*A*a*c*d^3*e^3*m^3 - 12*B*a*c*d^4*
e^2*m^2))/(e^6*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (x*(d + e*x)^m*(720*A*a^2*e^6 +
1044*A*a^2*e^6*m + 580*A*a^2*e^6*m^2 + 155*A*a^2*e^6*m^3 + 20*A*a^2*e^6*m^4 + A*a^2*e^6*m^5 + 342*B*a^2*d*e^5
*m^2 + 119*B*a^2*d*e^5*m^3 + 18*B*a^2*d*e^5*m^4 + B*a^2*d*e^5*m^5 - 144*A*c^2*d^4*e^2*m - 24*A*c^2*d^4*e^2*m^2
+ 360*B*a^2*d*e^5*m + 120*B*c^2*d^5*e*m - 480*A*a*c*d^2*e^4*m + 360*B*a*c*d^3*e^3*m - 296*A*a*c*d^2*e^4*m^2 -
60*A*a*c*d^2*e^4*m^3 - 4*A*a*c*d^2*e^4*m^4 + 132*B*a*c*d^3*e^3*m^2 + 12*B*a*c*d^3*e^3*m^3))/(e^6*(1764*m + 16
24*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (x^2*(m + 1)*(d + e*x)^m*(360*B*a^2*e^4 + 342*B*a^2*e^4*m
- 60*B*c^2*d^4*m + 119*B*a^2*e^4*m^2 + 18*B*a^2*e^4*m^3 + B*a^2*e^4*m^4 + 12*A*c^2*d^3*e*m^2 + 72*A*c^2*d^3*e*
m + 148*A*a*c*d*e^3*m^2 + 30*A*a*c*d*e^3*m^3 + 2*A*a*c*d*e^3*m^4 - 180*B*a*c*d^2*e^2*m - 66*B*a*c*d^2*e^2*m^2
- 6*B*a*c*d^2*e^2*m^3 + 240*A*a*c*d*e^3*m))/(e^4*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))
+ (B*c^2*x^6*(d + e*x)^m*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(1764*m + 1624*m^2 + 735*m^3 + 175*
m^4 + 21*m^5 + m^6 + 720) + (c^2*x^5*(d + e*x)^m*(6*A*e + A*e*m + B*d*m)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/
(e*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (c*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)
*(60*B*a*e^2 + 22*B*a*e^2*m - 5*B*c*d^2*m + 2*B*a*e^2*m^2 + 6*A*c*d*e*m + A*c*d*e*m^2))/(e^2*(1764*m + 1624*m^
2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (2*c*x^3*(d + e*x)^m*(3*m + m^2 + 2)*(120*A*a*e^3 + 74*A*a*e^3*
m + 10*B*c*d^3*m + 15*A*a*e^3*m^2 + A*a*e^3*m^3 + 30*B*a*d*e^2*m - 12*A*c*d^2*e*m + 11*B*a*d*e^2*m^2 + B*a*d*e
^2*m^3 - 2*A*c*d^2*e*m^2))/(e^3*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**m*(c*x**2+a)**2,x)
[Out]
Timed out
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